He should go to Washington more often. Gordon Brown may have been dreading his encounter with George Bush, knowing that every appearance Tony Blair made alongside the American president cost him votes by the crateload, but Monday's joint press conference at Camp David actually did Brown a favour. There was Bush, alternating between two of his least appealing personas: either frat-boy, mocking Nick Robinson's baldness, or cowboy, vowing his determination to track down the "cold-blooded killers" who do "evil". By turns he was condescending, telling Brown he had "proved his worthiness as a leader" during June's thwarted terror attacks, and rambling, eventually admitting that he was going on "too long". Next to Bush, Brown had only to read his script to look like a master communicator.

That's not all that went well. Brown wanted his Washington debut to look nothing like the Bush-Blair love-ins of the past, and he succeeded. Out went the groin-squeezingly tight jeans, in came the suits. No more "George", now it was Mr President. No more hugs between Laura and Cherie; this time the wives stayed at home. The backslaps were gone too, replaced by a shake of the hand. Every sign spelled out the same message: strictly business.

To the naked eye it may have seemed as if these superficial matters of body language and costume were all that had changed. But a close reading of Brown's words at Camp David, and indeed the content of his entire trip to the US, including his appearance at the United Nations yesterday, suggest otherwise. They indicate a shift not only in the so-called special relationship, but a deeper, strategic rethink in what Brown pointedly does not call "the war on terror".

That much was visible in the wide cracks of daylight the prime minister opened up between himself and the president. Yes, there were multiple avowals of shared purpose and common values. But while the president said the west confronted "an ideology of darkness", Brown declared that "terrorism is not a cause; it is a crime". That immediately denies the terrorist the dignity of an enemy and casts him instead as a mere criminal, to be hunted down chiefly by policework and intelligence. Noticeable too was Brown's desire to be specific: the conflict was not with "terror" - an abstract noun - but "al-Qaida-inspired terrorism".

The differences were even clearer on Iraq. Bush still speaks with righteous zeal for the mission, but Brown offered only the flat statement that Britain had "duties to discharge and responsibilities to keep." It was left to Bush to say that "Gordon Brown understands that failure in Iraq would be a disaster for the security of our countries", but the prime minister said nothing to confirm that statement of his views. Bush still refers to Iraq as the "central front" in the war on terror, but Brown named Afghanistan as the "frontline".

Above all, Brown put the US on notice for an eventual withdrawal of British troops from Iraq, dependent on the word of military commanders on the ground. Since that is the same formulation Bush himself uses, he could hardly object.

It's not quite Hugh Grant sticking it to Billy Bob Thornton in Love Actually, but this is about as far as a British prime minister could reasonably be expected to go in putting an American president at arms length. No, he didn't call Bush a deranged Texan gunslinger but nor did he return a single one of Bush's copious personal compliments. While the president tried to warm the air with talk of the "humorous Scotsman", Brown said their talks had been "full and frank" - icy diplomatese for a row. If breaking the clammy hug Blair and Bush shared was the goal, it certainly worked. A headline in yesterday's Washington Post declared of Brown: "More bulldog than poodle."

The Brown team would love to see that verdict repeated on every British front page, but this is about more than political positioning, fleeing an American president whose embrace has proved so toxic. It conceals a deep shift in what has been the central geopolitical question since September 2001.

Put simply, Brown sees the struggle against radical Islamism entirely differently from Bush, and therefore Blair. While their focus was on rogue regimes that posed a threat to the west, and the use of force to remove them, Brown sees a battle for the hearts and minds of the Muslim world. While the favoured comparison of the Bush-Blair era was the second world war against Hitler and fascism, Brown looks to the cold war with Soviet communism.

That conflict was long and, of course, had a military dimension extending to a nuclear arms race and proxy conflicts across the globe. But no less important, Brown believes, was the defeat of the intellectual case for communism. In an article in Monday's Washington Post, Brown recalled the educational and cultural links and exchanges between the west and those behind the iron curtain that steadily eroded the latter's faith in the Soviet system. This leads to an intriguing possibility, that Brown is advocating a process of systematic cultural engagement with the civil societies of the Arab and Muslim world - involving the "schools, universities, museums, churches, trade unions" whose engagement in the cold war he invokes so warmly.

Perhaps more importantly, Brown wants the west to regain the moral high ground he believes is essential to victory in any ideological struggle. He's long been an advocate on Aids, poverty and debt. But now there's an added urgency. If the west is seen to be acting justly, then it will be that much harder for Osama bin Laden to rail against wicked western imperialism. This is the right context to judge Brown's activity at the UN yesterday. With more passion than he ever showed at Camp David, he called for a "coalition of conscience" to implement the millennium development goals and for the deployment of more than 19,000 blue-helmeted UN troops in Darfur, because he believes in those moves - but also because he reckons they will blunt al-Qaida's rhetorical blade. As he wrote in the Post: "We must expose the contrast between great objectives to tackle global poverty and honour human dignity and the evils of terrorists who would bomb and maim people ... indifferent to the very existence of human life."

This amounts to a new philosophy in the conflict against jihadism. Instead of simply installing new regimes in the Muslim world, it seeks to prove itself the moral superior of violent Islamism. That would have enormous implications, invalidating almost every aspect of the "war on terror" as it has so far been conducted, from Guantánamo to Abu Ghraib to the invasion of Iraq itself. (It might also count against some of Brown's own ideas, such as extending to 56 days the period of detention without charge.)

Artfully, Brown has so far brought Bush with him, winning his backing on Darfur, for example. But he will need other allies, which is why his praise at Camp David was for America rather than the Bush administration, and why he made a point of visiting congressional leaders on Capitol Hill. From now on, the special relationship will be with the US, not the Bush White House. If only because Brown knows that Bush will be gone in 18 months - and he has every intention of staying in office much longer.